n(4n-20)+(2n+20)=180

Solution for n(4n-20)+(2n+20)=180 equation:

Simplifying
n(4n + -20) + (2n + 20) = 180

Reorder the terms:
n(-20 + 4n) + (2n + 20) = 180
(-20 * n + 4n * n) + (2n + 20) = 180
(-20n + 4n2) + (2n + 20) = 180

Reorder the terms:
-20n + 4n2 + (20 + 2n) = 180

Remove parenthesis around (20 + 2n)
-20n + 4n2 + 20 + 2n = 180

Reorder the terms:
20 + -20n + 2n + 4n2 = 180

Combine like terms: -20n + 2n = -18n
20 + -18n + 4n2 = 180

Solving
20 + -18n + 4n2 = 180

Solving for variable 'n'.

Reorder the terms:
20 + -180 + -18n + 4n2 = 180 + -180

Combine like terms: 20 + -180 = -160
-160 + -18n + 4n2 = 180 + -180

Combine like terms: 180 + -180 = 0
-160 + -18n + 4n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-80 + -9n + 2n2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-80 + -9n + 2n2)' equal to zero and attempt to solve:

Simplifying
-80 + -9n + 2n2 = 0

Solving
-80 + -9n + 2n2 = 0

Begin completing the square.  Divide all terms by
2 the coefficient of the squared term: 

Divide each side by '2'.
-40 + -4.5n + n2 = 0

Move the constant term to the right:

Add '40' to each side of the equation.
-40 + -4.5n + 40 + n2 = 0 + 40

Reorder the terms:
-40 + 40 + -4.5n + n2 = 0 + 40

Combine like terms: -40 + 40 = 0
0 + -4.5n + n2 = 0 + 40
-4.5n + n2 = 0 + 40

Combine like terms: 0 + 40 = 40
-4.5n + n2 = 40

The n term is -4.5n.  Take half its coefficient (-2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
-4.5n + 5.0625 + n2 = 40 + 5.0625

Reorder the terms:
5.0625 + -4.5n + n2 = 40 + 5.0625

Combine like terms: 40 + 5.0625 = 45.0625
5.0625 + -4.5n + n2 = 45.0625

Factor a perfect square on the left side:
(n + -2.25)(n + -2.25) = 45.0625

Calculate the square root of the right side: 6.712860791

Break this problem into two subproblems by setting 
(n + -2.25) equal to 6.712860791 and -6.712860791.

Subproblem 1
n + -2.25 = 6.712860791

Simplifying
n + -2.25 = 6.712860791

Reorder the terms:
-2.25 + n = 6.712860791

Solving
-2.25 + n = 6.712860791

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '2.25' to each side of the equation.
-2.25 + 2.25 + n = 6.712860791 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + n = 6.712860791 + 2.25
n = 6.712860791 + 2.25

Combine like terms: 6.712860791 + 2.25 = 8.962860791
n = 8.962860791

Simplifying
n = 8.962860791

Subproblem 2
n + -2.25 = -6.712860791

Simplifying
n + -2.25 = -6.712860791

Reorder the terms:
-2.25 + n = -6.712860791

Solving
-2.25 + n = -6.712860791

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '2.25' to each side of the equation.
-2.25 + 2.25 + n = -6.712860791 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + n = -6.712860791 + 2.25
n = -6.712860791 + 2.25

Combine like terms: -6.712860791 + 2.25 = -4.462860791
n = -4.462860791

Simplifying
n = -4.462860791

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {8.962860791, -4.462860791}
Solution
n = {8.962860791, -4.462860791}